intro optics - ppt v1part 01

Nature of lightThe Nature of Light

How does light carry energy from one point to another?18th Century: 2 “competing” theories.

Newton: Particle (corpuscular) theory. Why? Rectilinear propagation no bending around corners.

Huygens: Wave Theory Why? Light beams can pass through each other without influence.

Geometric and wave opticsCourse divided into two “halves”:

Geometric Optics: based upon Newton’s idea. Propagation of light is governed by Fermat’s Principle.

Wave (Physical) Optics: based upon Hygens’ idea. Propagation of light is governed by Huygens’ Principle (wave motion).

Geometric opticsGeometric (Ray) Optics

Deals with the direction of propagation of light “rays”.

How to make a light ray?

Source

ApertureAperture

Light Ray

Light emitted by any point on the source can be thought of as being composed of a “bundle” of rays.

Reflection / refractionTwo important means by which a light ray direction can be changed:

Reflection:

Refraction:

Reflecting surface

Medium 1 (incident)

Medium 2 (transmitting)

Ray Optics

Possible path of a light ray through an optical system.

Fermat’s principleFermat’s Principle

Original formulation: Light travels from point A B along the path which takes the least time.

Medium 1

v1 Medium 2

v2

Medium 3

v3

Medium 4

v4

Medium 5

v5

Assumption: Light travels at different speeds in different media. This was postulated by Huygens and Newton and later verified by experiment.

The propagation speed in vacuum c

Fermat’s principle 2Fermat’s Principle

It's not quite right to call this the principle of least time. In the figure below, for example, we consider light emitted at the centre of an elliptical mirror. The four physically possible paths by which a ray can return to the center consist of two shortest-time paths and two longest-time paths !!

Fermat’s principle 3Fermat’s Principle

More precise formulation: Time taken for light to travel along its true or actual path from point A B is equal “in first order approximation” to the time taken along other (hypothetical) paths closely adjacent to the true path.

Optical path lengthHomogeneous medium: Light travels from point A B along a straight line path. The time taken is:

Optical path length (OPL): the OPL is proportional to the time taken:

Path length

Speed of light in the medium

Index of refractionIndex of Refraction (of a medium)

For materials we’ll be considering (transparent): n 1

By definition: nvacuum = 1

Note: nair 1.000293 nvacuum

Terminology: if medium 2 has refractive index n2 and medium 1 has refractive index n1 > n2 then medium 1 is “optically denser”.

Note: refractive index n is colour dependent (“dispersion”)

Law of reflection: Fermat (1)Law of Reflection by Fermat’s Principle

Hypothetical light paths from point A B by reflection. Which is the true path?

Reflecting surface (xz plane)

Law of reflection: Fermat (2)Law of Reflection by Fermat’s Principle

Set up the problem for solution by Fermat’s Principle:

The “family” of hypothetical light paths from point A B is determined by the various possible values of x.

Law of reflection: Fermat (3)Law of Reflection by Fermat’s Principle

Fermat’s Principle: The true path OPL is equal to the OPL of nearby (adjacent) hypothetical paths (to “first order” approximation).

Law of reflection: Fermat (4)Law of Reflection by Fermat’s Principle

Here OPLAB =OPLAB(x):

Law of reflection: Fermat (5)Law of Reflection by Fermat’s Principle

For a nearby path, the OPL is OPLAB(x+ ) for small .

When does OPLAB(x+ ) = OPLAB(x) (to first order)?

Recall from elementary calculus that:

Law of reflection: Fermat (6)Law of Reflection by Fermat’s Principle

Apply the definition of the 1st order derivative to OPLAB(x):

For small :

“first order approximation”

Law of reflection: Fermat (7)Law of Reflection by Fermat’s Principle

Thus OPLAB(x+ ) = OPLAB(x) (to first order) when the 1st order derivative of OPLAB(x) vanishes:

The value of x for which the 1st order derivative of OPLAB(x) vanishes will determine the true path of the light ray from A B.

Law of reflection: Fermat (8)Law of Reflection by Fermat’s Principle

Solve this equation:

Law of reflection: Fermat (9)Law of Reflection by Fermat’s Principle

To make the solution more obvious, introduce the angles i and r (the angles of incidence and reflection).

Law of reflection: Fermat (10)Law of Reflection by Fermat’s Principle

Law of Reflection (partially!)

Fermat least timeFermat’s Principle

Note: The condition of “least time” and of “nearly equal OPLs for nearby paths” are equivalent.

The 2nd condition is more meaningful (later!).

Law of Refraction by Fermat’s principle 1

Law of Refraction by Fermat’s Principle

Points A and B are separated by an interface between two different media.

Incident Medium

Transmitting Medium

Interface

Possible (hypothetical) paths connecting points A and B.

Which is the true path?

Law of Refraction by Fermat’s principle 2

Law of Refraction by Fermat’s Principle

Set up the problem for solution by Fermat’s Principle.

Law of Refraction by Fermat’s principle 3

Law of Refraction by Fermat’s Principle

The OPL for the path A P B is thus:

Fermat’s Principle is satisfied for the condition:

Solving this equation for x will determine the true path.

Law of Refraction by Fermat’s principle 4

Law of Refraction by Fermat’s Principle

Define angles i (the angle of incidence ) and t (the angle of transmittance):

Law of Refraction by Fermat’s principle 5

Law of Refraction by Fermat’s Principle

Law of Refraction by Fermat’s principle 6

Law of Refraction by Fermat’s Principle

Fermat’s Principle gave:

Thus:

Snell’s Law of Refraction (partial)

Summary: Law of ReflectionSummary: Law of Reflection

Normal to reflecting surface at P.

Incident ray: Reflected ray

Vectors lie in the same plane, “plane of incidence”

Summary: Law of Reflection (alt)

Summary: Law of Reflection (alt)

Incident ray:

Reflected ray

Consider a light ray emanating from point O (unit vector r1) and reflected at point P to arrive at Q (unit vector r2). The reflecting surface is the xy plane.

Reflection at the xy plane gives the transformation of the incident ray unit vector to the reflected ray unit vector:

Application: Corner CubeApplication: Corner Cube (Retro) Reflector

Incident ray

Reflected ray

Unit vector r1 gives the incident light ray direction and unit vector r2 gives the reflected ray direction. Reflection at the 3 orthogonal surfaces transforms the incident ray unit vector to a reflected ray unit vector pointing in exactly the opposite direction.

Corner Cube: Three mutually orthogonal reflecting surfaces.

Summary: Law of RefractionSummary: (Snell’s) Law of Refraction

Normal to the interface at P.

Transmitted ray

Vectors lie in the same plane, “plane of incidence”

Reversibility Reversibility Principle

Fermat’s Principle gives the light path from A B independent of the direction in which light travels along this path. In other words, Fermat’s Principle applied to light travelling from A B gives the same path when applied to light travelling from B A

Uniform Medium Uniform Medium

In applying Fermat’s Principle to reflection and refraction, we assumed straight line propagation in a uniform medium. This can be proved using Fermat’s Principle (calculus of variations).

Uniform medium (constant n)

Non-uniform medium (non-constant n)